Finite Group Databasebeta

← #62544

Subgroups — #62544

Order 256.

892 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3509C211yesmaximal, characteristic, complemented
2#3508C211yesmaximal, characteristic
3#3371C214yesmaximal, complemented
4#3323C214yesmaximal, complemented
5#3317C214yesmaximal, complemented
6#3311C214yesmaximal, complemented
7#3231C211yesmaximal, characteristic, complemented
8#3228C211yesmaximal, characteristic, complemented
9#3227C211yesmaximal, characteristic, complemented
10#3210C211yesmaximal, characteristic, complemented
11#3210C211yesmaximal, characteristic, complemented
12#3208C211yesmaximal, characteristic, complemented
13#3185C211yesmaximal, characteristic, complemented
14#3182C211yesmaximal, characteristic, complemented
15#3167C211yesmaximal, characteristic, complemented
16#3141C211yesmaximal, characteristic, complemented
17#3138C211yesmaximal, characteristic, complemented
18#3131C211yesmaximal, characteristic, complemented
19#2965C211yesmaximal, characteristic, complemented
20#559—21no—
21#559C2212yes—
22#557—21no—
23#557C2212yescomplemented
24#556—21no—
25#556C2212yes—
26#552—21no—
27#552—21no—
28#552C2212yes—
29#552C2212yescomplemented
30#550—21no—
31#550C2212yes—
32#546—21no—
33#546—21no—
34#546C2212yes—
35#546C2212yes—
36#541—21no—
37#541—21no—
38#540—21no—
39#540—21no—
40#536—21no—
41#535—21no—
42#533C2211yescharacteristic
43#533C2211yescharacteristic
44#533C2211yescharacteristic
45#533C2211yescharacteristic, complemented
46#533C2211yescharacteristic, complemented
47#533C2211yescharacteristic, complemented
48#532C2211yescharacteristic
49#532C2211yescharacteristic
50#532C2211yescharacteristic
51#532C2211yescharacteristic
52C2×C4⋊Q8C2211yescharacteristic, complemented
53C2×#127C2211yescharacteristic
54#519—21no—
55#518—21no—
56C4×#41C2211yescharacteristic
57#500C2211yescharacteristic, complemented
58#498C2211yescharacteristic, complemented
59#498C2211yescharacteristic, complemented
60#494C2211yescharacteristic, complemented
61#492C2211yescharacteristic
62#492C2211yescharacteristic
63#490C2214yes—
64#489C2214yescomplemented
65#488C2214yescomplemented
66#487C2211yescharacteristic, complemented
67#487C2211yescharacteristic, complemented
68#487C2211yescharacteristic, complemented
69#487C2211yescharacteristic, complemented
70#486C2211yescharacteristic
71#486C2211yescharacteristic
72#486C2211yescharacteristic
73#486C2211yescharacteristic
74#486C2211yescharacteristic
75#486C2211yescharacteristic
76#485C2214yes—
77#482C2214yescomplemented
78#481C2214yescomplemented
79#479C2214yescomplemented
80#478C2214yescomplemented
81#476C2214yescomplemented
82#470C2214yes—
83#468C2214yescomplemented
84#465C2214yescomplemented
85#464C2214yescomplemented
86#464C2214yescomplemented
87#460C2214yes—
88#452C2214yescomplemented
89#452C2214yescomplemented
90#450C2214yes—
91#446C2212yescomplemented
92#441C2214yescomplemented
93#440C2214yescomplemented
94C4×SD16C2214yescomplemented
95#436C2212yes—
96#435C2212yes—
97#435C2212yescomplemented
98#431C2211yescharacteristic
99#431C2211yescharacteristic, complemented
100#431C2211yescharacteristic, complemented
101#418C2214yes—
102#417C2214yescomplemented
103#416C2214yescomplemented
104C4×M16C2211yescharacteristic
105C4×M16C2211yescharacteristic, complemented
106C2×C4×C8C2211yescharacteristic, complemented
107C2×#41—21no—
108C2×#41—21no—
109C2×#41D411yescharacteristic
110C2×#41C2312yes—
111C2×#41C2312yes—
112C22×Q8—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4C2312yes—
116C2×Q16C2314yes—
117C2×SD16C2314yes—
118C2×SD16C2314yes—
119C2×M16C2312yes—
120C2×M16C2312yes—
121C22×C8C2312yes—
122C4⋊Q8—22no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C2311yescharacteristic
126C4⋊Q8C2311yescharacteristic
127C4⋊Q8C2311yescharacteristic
128C4⋊Q8C2311yescharacteristic
129C4⋊Q8C2312yes—
130C4⋊Q8C2311yescharacteristic
131C4⋊Q8C2311yescharacteristic
132#127—22no—
133#127—21no—
134#127—21no—
135#127C2311yescharacteristic
136#127C2311yescharacteristic
137#127C2312yes—
138#126—22no—
139#126—22no—
140#126—22no—
141#126—22no—
142#125—22no—
143#125—22no—
144#125—22no—
145#125—22no—
146#125—21no—
147#125—21no—
148#125—21no—
149#125—21no—
150#125C2312yes—
151#125C2312yes—
152#124—22no—
153#124—22no—
154#124—21no—
155#124—21no—
156#124C2312yes—
157#124C2312yes—
158#123—22no—
159#123—22no—
160#123—22no—
161#123—22no—
162#123—22no—
163#123—22no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8C2312yes—
173C22⋊Q8C2312yes—
174C22⋊Q8C2312yes—
175C22⋊Q8C2312yes—
176#121—21no—
177#121—21no—
178#121—21no—
179#121—21no—
180#121—21no—
181#121—21no—
182#121—22no—
183#121—22no—
184#121—22no—
185#121—22no—
186#121C2312yes—
187#121C2312yes—
188#120—22no—
189#120—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8D411yescharacteristic
201C4×Q8C2312yes—
202C4×Q8C2312yes—
203C4×Q8C2312yes—
204C4×Q8C2312yes—
205C4×Q8C2311yescharacteristic
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4D411yescharacteristic
215C4×D4D411yescharacteristic
216C4×D4D411yescharacteristic
217C4×D4C2312yes—
218C4×D4C2312yes—
219C4×D4C2311yescharacteristic
220C4×D4C2311yescharacteristic
221C4×D4C2311yescharacteristic
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—22no—
227#117—22no—
228#117D411yescharacteristic
229#117D411yescharacteristic
230#117D411yescharacteristic
231#117C2312yes—
232#117C2312yes—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C2312yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C42C2311yescharacteristic
238C2×C42C2311yescharacteristic
239C2×C42C2311yescharacteristic
240#107C2314yes—
241#106C2314yes—
242#106C2314yes—
243C4⋊C8C2312yes—
244C4⋊C8C2312yes—
245C4⋊C8C2312yes—
246#103C2314yes—
247#103C2314yes—
248#103C2314yes—
249#103C2314yes—
250#102C2314yes—
251#102C2314yes—
252#102C2314yes—
253#102C2314yes—
254#102C2314yes—
255#102C2314yes—
256C22⋊C8C2312yes—
257C22⋊C8C2312yes—
258C22⋊C8C2312yes—
259#97C2312yes—
260#97C2312yes—
261C4×C8C2311yescharacteristic
262C4×C8C2311yescharacteristic
263C4×C8C2312yes—
264C4×C8C2311yescharacteristic
265C4×C8C2311yescharacteristic
266C24—22no—
267#41—42no—
268#41—44no—
269#41—44no—
270#41—82no—
271#41—82no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—44no—
278C2×Q8C2×D411yescharacteristic
279C2×Q8C2412yes—
280C2×Q8C2412yes—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—82no—
295C2×D4C2×D411yescharacteristic
296C2×D4C2×D411yescharacteristic
297C2×D4C2×D411yescharacteristic
298C2×D4C2412yes—
299C2×D4C2412yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4C2×D411yescharacteristic
306C22×C4C2×D411yescharacteristic
307C22×C4C2×D411yescharacteristic
308C22×C4C2×D411yescharacteristic
309C22×C4C2×D411yescharacteristic
310C22×C4C2×D411yescharacteristic
311C22×C4C2411yescharacteristic
312C22×C4C2411yescharacteristic
313C22×C4C2411yescharacteristic
314Q16—44no—
315SD16—44no—
316SD16—44no—
317M16—42no—
318M16—42no—
319C2×C8—42no—
320C2×C8C2412yes—
321C2×C8C2412yes—
322C2×C8C2412yes—
323C2×C8C2412yes—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4C2×D411yescharacteristic
347C4⋊C4C2×D411yescharacteristic
348C4⋊C4C2×D411yescharacteristic
349C4⋊C4C2×D411yescharacteristic
350C4⋊C4C2×D411yescharacteristic
351C4⋊C4C2×D411yescharacteristic
352C4⋊C4C2412yes—
353C4⋊C4C2412yes—
354C4⋊C4C2412yes—
355C4⋊C4C2412yes—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4C2×D411yescharacteristic
371C22⋊C4C2×D411yescharacteristic
372C22⋊C4C2×D411yescharacteristic
373C22⋊C4C2×D411yescharacteristic
374C22⋊C4C2×D411yescharacteristic
375C22⋊C4C2×D411yescharacteristic
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42C2×D411yescharacteristic
383C42C2×D411yescharacteristic
384C42C2×D411yescharacteristic
385C42C2×D411yescharacteristic
386C42C2×D411yescharacteristic
387C42C2×D411yescharacteristic
388C42C2411yescharacteristic
389C42C2411yescharacteristic
390C42C2411yescharacteristic
391C42C2411yescharacteristic
392C23—22no—
393C23—22no—
394C23—44no—
395C23—82no—
396C23—82no—
397C23C22×D411yescharacteristic
398C23C22×D411yescharacteristic
399C23C22×D411yescharacteristic
400Q8—24no—
401Q8—24no—
402Q8—41no—
403Q8—44no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407Q8—42no—
408D4—24no—
409D4—24no—
410D4—41no—
411D4—41no—
412D4—41no—
413D4—44no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—42no—
418D4—82no—
419D4—82no—
420D4—82no—
421D4—82no—
422D4—82no—
423D4—82no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—44no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4C22×D411yescharacteristic
445C2×C4C22×D411yescharacteristic
446C2×C4C22×D411yescharacteristic
447C2×C4C22×D411yescharacteristic
448C2×C4C22×D411yescharacteristic
449C2×C4C22×D411yescharacteristic
450C2×C4C22×D411yescharacteristic
451C2×C4C22×D411yescharacteristic
452C2×C4C22×D411yescharacteristic
453C2×C4C22×D411yescharacteristic
454C2×C4C22×D411yescharacteristic
455C2×C4C2511yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8—22no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—164no—
476C22C23×D411yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4#57512yes—
493C4#57712yes—
494C4#58512yes—
495C2—41no—
496C2—41no—
497C2—41no—
498C2—82no—
499C2—82no—
500C2#350011yescharacteristic
501C2C2×#57511yescharacteristic
502C2C2×#57711yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3508C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3509C2semidirect
3. #3371C2semidirect
4. #3323C2semidirect
5. #3317C2semidirect
6. #3311C2semidirect
7. #3231C2semidirect
8. #3228C2semidirect
9. #3227C2semidirect
10. #3210C2semidirect
11. #3210C2semidirect
12. #3208C2semidirect
13. #3185C2semidirect
14. #3182C2semidirect
15. #3167C2semidirect
16. #3141C2semidirect
17. #3138C2semidirect
18. #3131C2semidirect
19. #2965C2semidirect
23. #557C22semidirect
29. #552C22semidirect
45. #533C22semidirect
46. #533C22semidirect
47. #533C22semidirect
52. C2×C4⋊Q8C22semidirect
57. #500C22semidirect
58. #498C22semidirect
59. #498C22semidirect
60. #494C22semidirect
64. #489C22semidirect
65. #488C22semidirect
66. #487C22semidirect
67. #487C22semidirect
68. #487C22semidirect
69. #487C22semidirect
77. #482C22semidirect
78. #481C22semidirect
79. #479C22semidirect
80. #478C22semidirect
81. #476C22semidirect
83. #468C22semidirect
84. #465C22semidirect
85. #464C22semidirect
86. #464C22semidirect
88. #452C22semidirect
89. #452C22semidirect
91. #446C22semidirect
92. #441C22semidirect
93. #440C22semidirect
94. C4×SD16C22semidirect
97. #435C22semidirect
99. #431C22semidirect
100. #431C22semidirect
102. #417C22semidirect
103. #416C22semidirect
105. C4×M16C22semidirect
106. C2×C4×C8C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#35092
#35082
#33712
#33232
#33172
#33112
#32312
#32282
#32272
#32102
#32082
#31852
#31822
#31672
#31412
#31382
#31312
#29652